## Finite Elements: Theory, Fast Solvers, and Applications in Solid MechanicsThis is a thoroughly revised version of the successful first edition. In addition to up-dating the existing text, the author has added new material that will prove useful for research or application of the finite element method. The most important applications of this method receive an in-depth treatment in this book. This textbook is ideal for graduate students without any particular background in differential equations, but who require an introduction to finite element methods. The author includes a chapter to bridge the gap between mathematics and engineering. |

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### Contents

Chapter I | 1 |

The Maximum Principle | 12 |

A Convergence Theory for Difference Methods | 22 |

Sobolev Spaces | 28 |

Variational Formulation of Elliptic BoundaryValue Problems | 34 |

The Neumann BoundaryValue Problem A Trace Theorem | 44 |

The RitzGalerkin Method and Some Finite Elements | 53 |

Some Standard Finite Elements | 60 |

Chapter IV | 177 |

Gradient Methods | 187 |

Preconditioning | 201 |

Saddle Point Problems | 212 |

Multigrid Methods for Variational Problems | 217 |

Convergence of Multigrid Methods | 228 |

Convergence for Several Levels | 239 |

Nested Iteration | 246 |

Approximation Properties | 76 |

Error Bounds for Elliptic Problems of Second Order | 89 |

Computational Considerations | 97 |

Chapter III | 105 |

Isoparametric Elements | 117 |

Saddle Point Problems | 129 |

Mixed Methods for the Poisson Equation | 143 |

The Stokes Equation | 154 |

A Posteriori Error Estimates | 169 |

Multigrid Analysis via Space Decomposition | 252 |

Nonlinear Problems | 263 |

Chapter VI | 269 |

Hyperelastic Materials | 281 |

Membranes | 304 |

The MindlinReissner Plate | 324 |

337 | |

### Other editions - View all

Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics Dietrich Braess No preview available - 2007 |

Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics Dietrich Braess No preview available - 2001 |

### Common terms and phrases

algorithm assertion assume bilinear form boundary conditions boundary-value problem bounded Braess Bramble-Hilbert lemma Brezzi Cauchy Cauchy-Schwarz inequality classical solution coarse-grid correction compute condition number conjugate convergence rate corresponding decomposition defined derivatives differential equations Dirichlet Dirichlet boundary conditions discrete domain edge elliptic energy norm equivalent error finite element approximation finite element solution finite element spaces five-point stencil follows formula Gauss-Seidel method grad gradient method Green's formula grid Hilbert space hypotheses implies inequality inf-sup condition interpolation iterative methods Jacobi method Lemma matrix maximum principle mixed methods multigrid method nodes nonconforming nonlinear operator orthogonal parameter penalty term piecewise Poisson equation positive definite preconditioning proof quadratic saddle point problem satisfies shape-regular Show smoothing steps Sobolev spaces Stokes problem subspace Suppose symmetric tensor Theorem theory triangle triangular elements V-cycle variational problem vector